Automatic loop-shaping of QFT controllers via linear programming

In this paper we focus on the following loop-shaping problem: Given a nomin al plant and QFT bounds, synthesize a controller that achieves closed-loop stability, satisfies the QFT bounds and has minimum high-frequency gain. Th e usual approach to this problem involves loop shaping in the frequency dom ain by manipulating the poles and zeroes of the aided design tools, proceed s by trialand error, and its success often depends heavily on nominal loop transfer function. This process now aided by recently-developed computer th e experience of the loop-shaper. Thus, for the novice and first-time QFT us ers, there is a genuine need for an. automatic loop-shaping tool to generat e a first-cut solution. Clearly, such all automatic process must involve so me sort of optimization and, while recent results on convex optimization ha ve found fruitful applications in other areas of control design, their imme diate usage here is precluded by the inherent nonconvexity of QFT bounds. A lternatively, these QFT bounds cart be over-bounded by convex sets, as done in some recent approaches to automatic loop-shaping, but this conservatism can have a strong and adverse effect on meeting the original design specif ications. With this in mind, we approach the automatic loop-shaping problem by first stating conditions under which QFT bounds can be dealt with in a non-conservative fashion using linear inequalities. We will argue that for a first-cut design, these conditions are often satisfied in the most critic al frequencies of loop-shaping and are violated infrequency bands where app roximation leads to negligible conservatism in the control design. These re sults immediately lead to an automated loop-shaping algorithm involving onl y linear programming techniques, which we illustrate via an example.